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Besides the unitarity and symmetry requirements for a multi-resonancescattering amplitude, several other natural conditions can easily excludeunrealistic proposals. In particular, the behaviour of singularities under thevariation of model parameters yields important information. We discuss howresonance poles should move in the complex-energy plane when coupling constantsand masses are varied, how resonances above threshold can turn into boundstates below threshold and how the light-quark spectrum can be turned into thespectrum of heavy quarks, with one and the same analytic expression for thescattering amplitude. Moreover, it is shown that perturbative approximationsusually do not satisfy those natural conditions.

General unquenching properties of two-meson scattering and production amplitudes